From a group of 7 boys and 6 girls, a team of 5 is to be selected so that there are at least 3 boys in the team. In how many ways can this be done?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 756
"At least 3 boys" means the team has 3, 4 or 5 boys. Count each case separately (choose the boys and the girls, multiply) and add the cases.
3 boys and 2 girls: \(^7C_3\times{}^6C_2=35\times15=525\)
4 boys and 1 girl: \(^7C_4\times{}^6C_1=35\times6=210\)
5 boys and no girl: \(^7C_5=21\)
Total \(=525+210+21=756\)
Why the other options are incorrect
Option (1) 645 — Incorrect. It matches no set of the three cases; the cases add to 756.
Option (2) 734 — Incorrect. It falls short of 756; some teams with 3, 4 or 5 boys are left out.
Option (4) 612 — Incorrect. It is below 756 as well, so not every allowed team is counted.